Origin of Giant Phonon Magnetic Moment in Orbital Seebeck Effect: a Heisenberg-type L-L Coupling

Abstract

Inspired by the recent observation of the orbital Seebeck effect in alpha-quartz, we identify an intrinsic amplification mechanism for thermally generated phonon angular momentum and phonon magnetic moment in chiral insulators. We propose a Heisenberg-type long-range coupling between phonon angular momenta, referred to here as L-L coupling, which opens a self-consistent feedback channel and strongly enhances the bare thermal response within linear response. Our calculations reveal a pronounced temperature- and size-dependent amplification, dominated by the off-diagonal channel, with the total phonon angular momentum enhanced by up to nearly two orders of magnitude as the system approaches the threshold from below. These findings suggest that L-L coupling may provide a microscopic origin of giant phonon magnetic moment the recently observed orbital Seebeck effect in alpha-quartz.

Figures (3)

• Figure 1: Schematic illustration of the self-consistent amplification mechanism for thermally generated PAM in a symmetric-top-like chiral insulator. Laser heating establishes an axial temperature gradient along the principal $\mathrm{c}$ axis of the sample, which induces local PAM. Through L-L coupling, local rotational motions become correlated and feed back positively on response coefficient, leading to an enhanced PAM.
• Figure 2: (a) Crystal structure of $\alpha$-quartz, highlighting the helical arrangement of SiO$_{4}$ tetrahedra along the $c$-axis. (b) Corresponding top view of the quartz structure. (c) First Brillouin zone and the high-symmetry points (blue). The relations among the real-space lattice vectors ($\mathbf{a,b,c}$; black), reciprocal-space lattice vectors ($\mathbf{a^{*},b^{*},c^{*}}$; green), and Cartesian axes ($x,y,z$; purple) are indicated. (d) Phonon dispersion color coded by phonon polarization $l_{\mathbf{k}\sigma}^{z}$. (e) Atomic species-projected phonon polarization of the first acoustic branch along the $\Gamma$-$\mathrm{A}$ direction. (f) Real-space visualization of the atomic rotational motion for the phonon mode at point $B$ marked in (d). The initial phase (red dots), motion trajectories (purple curves), and motion directions (blue triangles) are indicated.
• Figure 3: (a) Schematic illustration of the cylindrical sample under an axial temperature gradient $\nabla T$, which induces a PAM. (b) Temperature-size phase diagram of the L-L coefficient $\gamma^{\mathrm{LL}}$. The contours at 0.1, 0.5, and 1.0 are shown, with the $\gamma^{\mathrm{LL}} = 1$ boundary separating the subcritical and supercritical regimes. The horizontal dashed line marks the representative cut at $r=20$ nm. (c) Temperature dependence of $\gamma^{\mathrm{LL}}$ at 20 nm, together with its diagonal and off-diagonal contributions. The inset shows the corresponding amplification factor $\eta=(1-\gamma^{\mathrm{LL}})^{-1}$. (d, e) Brillouin-zone-resolved contributions to the diagonal and off-diagonal parts at $T=300$ K and $r=20$ nm, shown on selected $k_{z}$ slices. The color scales are chosen independently for clarity. (f) Total PAM density as a function of temperature without and with LL coupling for different cylinder radii, showing the strong size-dependent enhancement